Moving actuator surfaces : A new concept for wind turbine aerodynamic analysis

Transcription

1 Moving atuator surfaes : A new onept for wind turbine aerodynami analysis Christian Masson and Christophe Sibuet Watters 1 Department of Mehanial Engineering Éole de Tehnologie Supérieure 1100 Notre Dame Ouest, Montréal H3C 1K3 (Canada) Phone number: , hristian.masson@etsmtl.a Abstrat The atuator disk is a onept often used in wind turbine aerodynamis, where the ation of the turbine on the flow is eraged over time and spae. This simplifiation stills retain suffiient physial information for wind turbine aerodynami design. Its limitations are essentially related to the impossibility to model the blade shed vortiity. This paper presents a more general approah that uses rotating atuator surfaes arrying veloity and pressure disontinuities whih are set from knowledge of the irulation around the model blade setions and using blade element analysis. The mathematis of rotating atuator surfaes, as well as an appliation to the TUDelft rotor are presented in this paper. The results are enouraging. Key words Wind Turbine Aerodynamis, Atuator Surfaes, CFD 1. Introdution The aurate predition of wind turbine aerodynamis is a problem that has reeived onsiderable attention sine the beginning of the aerodynami siene. Apparently, the art of prediting wind turbine performane has not reah a onsensus yet, as shown by the blind ode omparison lead by NREL [1] that reported important disrepanies between predited power urves for 15 types of omputer-based analysis. Espeially in operation where the blade is partly, or ompletely stalled, preditive models meet major diffiulties to ensure auray: reasons for this poor performane an be attributed to the model itself, or to the inputs provided to the model, like the airfoil aerodynamis harateristis. Amongst the soures of disrepanies, the way models handle blade tip vorties is reognized as a key issue by many workers in the field [,3,4]. Theoretially, only vortex [18] or full CFD [19] analyses naturally model the tip vorties and its interation with the flow. Atuator disk representation of the wind turbine suffers for lak of physial representation of the tip vorties and makes use of a tip fator to ompensate for their influene. This paper presents new advanes towards the development of a numerial method that models the flow around a wind turbine by surfaes arrying veloity disontinuities, with the objetive to improve preditions of blade tip vorties effets on performane. Previous papers he demonstrated the validity and pertinene of this method with appliations to problems hing analytial solutions and to the ase of the atuator disk [5,6] ompared to methods that represent the atuator disk by a set of external fores only [7,8,9,10], no spurious osillations of the flow parameters happen. First, mathematial and numerial aspets of the method are briefly reviewed. A ontrol-volume finite-element method is used to solve the flow in the non-inertial system of oordinates attahed to the blades. Several adaptations are neessary to ensure numerial onvergene and are presented. Comparisons between numerial results and experimental measurements are presented for the TUDelft experimental rotors [16].. Mathematial Aspets of Atuator Surfaes Theoretial aspets exposed in the next setions are taken from lassial onepts of inompressible fluid mehanis. The original idea presented regards the introdution of porous surfae of veloity and pressure disontinuities in the flow, and fous is made on the onsequenes of these disontinuities on the flow, as well as to their proper evaluations. A. Atuator Surfaes - Impliations In the ontext of invisid flow theory, the ation of a lifting devie an be represented by a set of vortiity distributions along surfaes (vortex sheets), and the flowfield is solely determined by this set. In this type of modelization, the vortiity surfaes are neessarily streamlines and hene an not be rossed by fluid. From a kinemati point of view, vortex sheets are however equivalent to surfaes of veloity disontinuities. The RE&PQJ, Vol. 1, No.6, Marh 008

2 general idea of the atuator surfae is to use one single, zero-thikness porous surfae to model the effet of a lifting devie on the flow. It is therefore interesting to study impliations due to suh singular surfae in the ontext of the Eulerian desription of the flow with veloity omponents (u,v,w) and pressure p measured in a lassial Cartesian (x,y,z) system of axes. Δp z= z P z y x Δv Δu Fig. 1: The surfae of disontinuities Fig. 1 presents a retangular surfae to be used to analyze a finite retangular wing aerodynamis. This surfae is flat and lies in a plane of onstant z=z P. It arries veloity disontinuities u and v as well as a pressure jump p. By onvention, all disontinuities measure the jump of parameter value aross the surfae from the lower side (z< z P ) to the upper side (z>z P ). In the general ase studied, the surfae is porous and an be rossed by fluid flow. The line integral of u along a blade setion is equal to the irulation Γ around this setion. For the singular surfae of Fig. 1, this would be mathematially expressed as 0 Δu. dx = Γ along a line( z = z ; y = y ) The ourrene of veloity disontinuities aross the surfae must respet, from vortiity onservation priniple, the following partial differential equation: ( Δv) ( Δu) x y (1) From loal ontrol volume analysis, Leler & Masson [11,1] he shown the following general results for atuator surfaes : At a given loation on a surfae of veloity disontinuity, the veloity disontinuity is neessarily tangent to the surfae, and provided that no energy is withdrawn from or added to the flow, then a system of fore attahed to the surfae of disontinuity naturally arises and the omponents of this external fore, per unit area are given by ( in the ase of the finite wing of Fig.1): f f f x y z = ρw = ρw Δu Δv = ρ( u Δu + v Δv) P e where u, v, w are the omponents of the mean veloity at the surfae of disontinuity, given by: u v w = ( u + ( u + Δu)) / = ( v + ( v + Δv)) / = ( w + ( w + Δw)) / It is interesting to note that the salar produt of the mean veloity and the fore vetor is null, whih implies that the system of fore attahed to the surfae of disontinuity does not exert any work on the flow. A seond, diret onsequene of this system of fore is related to the normal omponent f z whih implies that there must exists a pressure jump at the atuator surfae, whose magnitude is exatly equal to f z. B. Blade-element theory for the finite wing β α i U Fig. : Cross setion of a wing with parameters of interest Fig. presents a sketh of a wing setion with inoming flow U. The aerodynami setion of hord length is pithed with an angle β towards the inoming flow. In the ase of an infinite wing, i.e. an airfoil, the fore produed by a uniform air flow would onsist in a lift fore perpendiular to the diretion of the flow. Finite wing theory however, teahes that the system of downwind trailed vortiity indues a downwash of the flow around the wing (a vertial, downward omponent of the flow at the wing position) measured by the indued angle of attak α i. Atual lift fore is therefore inlined and its projetion on the diretion of the inoming flow is lassially denoted D i, the indued drag. L is the lift of the wing being defined as the fore perpendiular to the inoming flow. The finite wing is modeled numerially as a flat surfae parallel to the inoming flow whih bears veloity disontinuities. x is the axis in the diretion of the inoming flow and y the axis in the spanwise diretion of the wing, as shown in Fig. 3. For a setion of the wing, the lift per unit length L is given by the equation: 1 L = ρ U C L L D i Γ RE&PQJ, Vol. 1, No.6, Marh 008

3 where ρ is the density of the air and C L is the lift oeffiient of the airfoil defining the wing setion, funtion of the relative angle of attak α=β-α i. Let Γ be the irulation of veloity around the airfoil, from the Kutta-Jukowski law, it is also possible to write: L = ρ U Γ Combining the last two equations, Γ may be rewritten as: 1 Γ = r t U C L P x P Fig. 3: Flat surfae modeling the tapered wing One the irulation Γ around an airfoil is known, the task onsists in distributing this irulation along the airfoil hord in the most appropriate way. The onstant distribution u= Γ/ is the simplest but performs poorly sine it sets rapid variations of the u field, hard to manage numerially without instabilities in the flow solution. Distributions respeting ontinuity of the u(x,y,z) are more appropriate. In this work, results will be obtained using a simple paraboli distribution: 6Γ Δ up = x ( ) 3 P xp where x P is the distane from the point P to the leading edge of the wing along the x-axis diretion, as shown in Fig.3. The distribution of v is dedued from Eq. (1) and the boundary ondition v =0 at x P =0: Δ v P 6Γ = y x 3 1 Γ 3 ( x x ) + ( 3x x ) 4 P P 3 P P y All results shown in the present work are performed using the above distributions of u and v. Sibuet Watters & Masson [1] he obtained detailed results for the ase of the finite tapered wing: For low wing aspet and taper ratios, the method has shown to yield good results of the indued drag and of the indued downwash veloity at the wing. Grid-independeny has been studied and shows that a minimum number of 1000 nodes on the surfae arrying disontinuities are needed. The proposed method has also shown to be a good tool to evaluate the relationship between inflow angle measured at some loation upstream, and loal angle of attak. y In this artile, the method is used to model wind turbine blades, modelled by flat surfaes in rotation. Adaptations to the finite wing under translation analysis lie only in the appliation of the blade element theory: loal angle of attak is evaluated from loal wind veloity (U is evaluated from the ombination of loal flow veloity and rotational veloity) and knowledge of blade geometry harateristis: pith and twist angles. 3. Numerial Implementation of the wind turbine model A. Presentation of the model To solve the set of partial differential equations desribing the flow evolution (the Nier-Stokes equations in their inompressible, steady-state form), the 3D Control-Volume Finite-Element Method (CVFEM) of Saabas & Baliga [13] is used with appropriate modifiations. The ode used is typial, in its formulation, of modern ommerial CFD software like CFX Ansys and Fluent. An in-house CVFEM ode is preferred here sine the flexibility of the ommerial software is not suffiient to implement the proposed mathematial model. For the analysis of two-blade rotors, onsidering symmetry of the problem, the domain of solution is a half hollow ylinder. The alulation domain is disretized using a strutured mesh and tetrahedral finite elements, whih themselves are further divided to build ontrol volumes (CVs) around eah domain node; diffusion fluxes are alulated assuming linear variation of flow harateristis within geometry elements defining the CV while onvetion fluxes are alulated using onvetion shemes. Visous diffusion and gradient of pression are evaluated at the ontrol surfaes using linear distributions of veloity and pressure within one tetrahedral element. Convetion fluxes however, are evaluated using the firstorder Mass-Averaged-weighted (MAW) sheme developed by Saabas & Baliga [13]. Assembling the ontributions of all tetrahedral element of one ontrol volume for mass and momentum balane builts the sets of disretized equations to solve numerially ; Mass fluxes are related to pressure gradients suh that the disretized pressure equations are elaborated from the mass onservation priniple. B. Reformulation of the mathematial problem The rotating system of oordinates, where the blades appear fixed, has been hosen to model the rotor. The governing equations are the Nier-Stokes equations (NS) in whih inertial (entrifugal and Coriolis) fores appear. When pressure and flow veloity omponents (expressed in the Cartesian oordinates and measured in the rotating system) are seleted as the unknowns to this problem, and the CVFEM formulation of Saabas & Baliga [13] is used to build the algebrai equations for these unknowns, numerous onvergene problems were arising, even for the simple ase of the advaning flow in RE&PQJ, Vol. 1, No.6, Marh 008

4 solid rotation without obstale (orresponding to the inlet ondition of the wind turbine problem). Investigations he determined that the onvetion sheme is in fat inappropriate to model adequately the onvetion fluxes speifially due to the rotating omponent of the veloity Ω r (Ω is the blade rotational speed and r is the distane to the axis of rotation). Suh onvergene problems would not arise in the ase of a CVFEM implemented to solve the NS equations written in the ylindrial oordinates. Commerial odes like Fluent or CFX Ansys he speifi implementations to treat flow analysis in rotating system. Unfortunately, details of their implementations ould not be found and ompared to the one adopted in this work and desribed in the rest of this setion. To resolve this onvergene problem, the relative veloity u as measured in the rotating system of referene is split in two parts: u = u Ωreθ () where u is the veloity vetor measuring the flow veloity in a fixed system of oordinates and Ωreθ is the rotational veloity vetor. u at every domain nodes onstitute the new unknowns of this problem. BLADE Ω Fig. 4: Parameters for wind turbine study ; the arrow around Ω indiates blade diretion of rotation (observer is upstream the blade) r This separation of the veloity in two terms is introdued in the NS equations: u( u. n) ds = pnds + Ω re Ω ρ ρ r ρ S S where the surfae integrals are taken upon a ontrol volume V of surfae S with n as the outward normal vetor to the surfae. The last two terms are inertial fores (entrifugal and Coriolis fores) due to the rotation of the referene system. e θ θ V e r u Following Eq.(), the onvetion term is split in three terms : ρu( u. n) ds = + ρu ρωre ρω r ( u. n) ds θ ( u e ( e θ. n) ds θ. n) ds (3) On the right-hand side of Eq.(3), the first term an readily be used to model the ontribution of onvetion to the disretized momentum onservation equation using u nodal values as unknowns, following the MAW sheme of Saabas & Baliga [13]. The seond and third terms on the right-hand side of Eq.(3) are treated as onstant soure terms in the numerial model when building the algebrai equations from momentum onservation in a ontrol volume. The sufae integral of the seond term is alulated at Gaussian points on the surfae and mass fluxes are evaluated using the MAW sheme of Saabas & Baliga [13]. It an be proven that the third term in the right-hand side of Eq.(3), ombined with the inertial fores, will result in ρω u a total volume fore given by, where Ω orresponds to the rotation vetor (with diretion along the axis of rotation). This volume fore is in some way, a Coriolis fore that is applied only on the nootating omponent of the veloity. While the deomposition presented by Eq.() may appear unjustified, there is a more physial way to obtain the same final form of the governing equations by applying momentum onservation expressed in the non-rotating, inertial system of referene over rotating volumes of ontrol (at rate Ω ). This reformulation of the governing equation has proven to be very effiient and has been thoroughly tested on the ase of the advaning flow in solid rotation without obstale; different boundary onditions and initial first guess of the solution he been used on this ase: in almost every simulation, the solution would rapidly onverge towards the trivial solution u = 0. C. Numerial Treatment of disontinuities Handling properly the ourrene of veloity and pressure disontinuities in the 3D CVFEM neessitates numerous modifiations in the numerial ode and the introdution of new soure terms in the disretized momentum equations. This subjet has been explained in previous ontributions (see [14]) and will not be further addressed in the present one. It should be noted that ommerial odes, at the present time, are unable to ope with veloity disontinuities, while some are designed to handle pressure jumps aross surfaes (as a model to porous media for example) RE&PQJ, Vol. 1, No.6, Marh 008

5 (6) Iterate Steps () to (5) until a onverged solution is reahed. J= J=1 X u (I,1,K)= u (I,NJ,K) v (I,1,K)= - v (I,NJ,K) w(i,1,k)= - w(i,nj,k) Y Z J=NJ Fig.5 : Mesh for the blade in rotation (only downwind part shown ; blade position is indiated using blue olor) D. Boundary onditions As shown in Fig. 5, the solution domain is a half hollow ylinder. Depending on the flow variable solved, periodi or aperiodi boundary onditions are set at the border of the mesh. As shown in the zoom of Fig. 5, the strutured mesh is omposed of radial lines extending from an inner radial position (set small ompared to the blade dimensions) to an outer position far from the blade. Let I,J and K be the indies pointing a speifi node of the strutured mesh, then the previously mentioned radial lines orrespond to J=Cte lines, and providing that u, v and w are the flow omponents in the x, y and z axis as shown in Fig. 5, then periodiity is expeted for u, and aperiodiity for v and w omponents between surfaes J=1 and J=NJ as summarized by the formulas in Fig. 5. Consequently, boundary onditions used in the problem under study are either of the Dirihlet (fixed values) type or Neumann type (see Ref.[13] for a deeper knowledge of outflow onditions), exept at sides of the half ylinder, where original aperiodi onditions he been implemented. Regarding boundary onditions for the pressure equation, the pressure is set onstant at one node of the domain, and only the downstream surfae is treated as an outflow boundary. 4. Results In this paper, the aerodynamis of an experimental rotor designed at Delft Tehnial University (TUDelft) is thoroughly studied. The TUDelft rotor is a two-blade horizontal-axis wind turbine hing a diameter D of 1.m that rotates (for the ases studied in the artile) at 700 rpm. A NACA001 profile is used on the lifting part of the blade from r/r=0.3 up to the tip, and the blade has onstant hord length of 0.08m with linear twist distribution. Detailed information regarding the wind turbine and the researh program of TUDelft an be found in the works of Sant [15] and Haans et al.[16]. The blade an be pithed at different angles. In the present study, blade tip pith angles of 0, and 5 degrees are used. Detailed inflow and near wake measurements he been performed using hot film anemometry. A. Calulation set-up The former studies for the finite wing disretization [1] he enable to raise a number of reommendations on the onstrution of alulation domain and mesh for atuator surfae studies. Following these reommendations, a mesh of 01*65*46 nodes is used along axial (I), azimuthal (J) and radial (K) diretions to disretize the domain of total length 3.5D, inner radius D, outer radius 5D where D=1.m is the rotor diameter. Mesh is refined around the atuator surfae loation, whih is loated at 0.3D upstream of the outflow boundary. As shown in Fig. 6 whih presents the nodes distribution in the I=Cte plane where the atuator surfae lies, 17*0 nodes are used to represent the atuator surfae: E. Overall solution algorithm The solution algorithm for the proposed moving atuator surfaes model is the following: (1) Initialize flow veloities to undisturbed values, i.e. set u = 0 at every node in the flow; () Calulate the spanwise distribution of effetive angle of attak α from knowledge of loal flow veloities at the wing and blade-element theory; (3) Calulate C L oeffiient aross the wing using tables and α distribution; (4) Calulate distribution of irulation Γ and set u and v using equations of Setion B; (5) Solve for the flow (i.e. for u and pressure p) using the presribed distributions of veloity disontinuities u and v; Fig. 6 : Axial mesh plane at atuator surfae The loation of inflow and outflow boundaries relative to the atuator surfae has been studied to ensure independeny of results, together with the influene of the spaing between axial mesh planes in the viinity of the atuator surfae. The mesh seleted to produe the results presented in this paper is, in terms of total number of nodes, a good ompromise between auray (in terms of mesh independeny) and omputing time RE&PQJ, Vol. 1, No.6, Marh 008

6 Relative flow veloities and angles for the blade element analysis are all evaluated at the middle of the blade, i.e. the vertial line that splits the atuator surfae in two equal parts. D oeffiients of lift and drag are taken from [17] and for a Reynolds number of As explained in the presentation of the atuator surfae, no tip fator model is needed and for the setting of disontinuities at the atuator surfae, only lift harateristis are needed. A more elaborated model (see [15] for a good review of existing models) would be welome to orret the setion aerodynamis fores for the effets of rotation (due to phenomenon like stall delay). Here, however, only D harateristis he been used. For all omputations, an artifiial visosity was employed to ensure stability during the onvergene proess ; it was verified that the level used did not influene the rotor inflow and near-wake results. B. Power and Thrust Curves First results presented in Figs. 7 and 8 are the omputed oeffiient of power and thrust against tip-speed ratio for tip-pith angles of 0 º, º and 5 º. to experiments only for the º tip-pith angle ase, otherwise there an be strong disordane. C. λ=8 ase : Components of veloity To give an insight into the model results, Figs. 9 to 11 present the veloity omponents (in m/s) alulated by the model for λ=8, orresponding to a uniform inoming veloity of 5.5 m/s with the rotor rotating at 700 rpm. The three figures are isoontours of the three ylindrial (as defined by the system of oordinates of Fig. 5) veloity omponents of u taken either in the plane enlosing the rotor swept surfae (Fig. 9) or in the axial plane that splits the blade in two halves, i.e. θ=π/ (Figs. 10 and 11). In the last two figures, loation of the atuator surfae is indiated by a straight bold line. Sine it represents the non-rotating omponents of veloity, these pitures represent the flow state when a blade reahes the upside top position. It is interesting to note that the model handles well (1) the effet of rotation, () the veloity disontinuities, (3) the existene of strong vertial strutures emanating from both the tip and root areas of the rotor, and (4) the indution of the atuator surfae on the flow upstream the rotor. Fig. 11 is partiularly striking and shows that the model an be used to trae tip vorties trajetories. Fig.7 : Power oeffiient as a funtion of tip-speed ratio Fig.9 : Axial omponent of u, λ=8 Fig. 8 : Thrust oeffiient as a funtion of tip-speed ratio Power is alulated from blade element theory using loal flow veloity and D lift and drag harateristis. Thrust is alulated from surfae integration of the pressure disontinuity over the atuator surfae. The numerial model results are ompared with experimental measurements. Agreement with experiment is muh better for the thrust than for the power oeffiient. Coeffiient of power values found by the model are lose Fig. 10 : Tangential omponent of u, λ=8, flow is from left to right RE&PQJ, Vol. 1, No.6, Marh 008

7 Fig. 11 : Radial omponent of left to right u, λ=8, flow is from For the partiular ase of λ=8, detailed experimental measurements he been obtained by TUDelft [15,16] using hot-film anemometry, phase eraging and original reombination methods to derive the 3D omponents of veloity. These measurements were taken both in the inflow and near wake region of the turbine along planes parallel to the rotor swept surfae. Figs. 1, 13 and 14 present omparisons at planes loated 0.03D, 0.05D and 0.075D (orresponding to 3.5, 6 and 9 m) downstream the rotor between the numerial results and the experimental measurements for the axial, tangential and radial omponents of the non-rotating veloity (in m/s). In eah figure, three radial loations are presented : r/r=0.4 (blade root area), r/r=0.7 (mid-blade area) and r/r=0.9 (blade tip area). Components of veloity are plot against azimuth θ. It an be observed that signifiant bias our between experimental measurements and model outputs for the axial and radial omponents. However, its is remarkable that the shape of azimuthal distributions of veloity omponents are very idential between the model and the measurements. Furthermore, it may be noted that the bias between experimental and modelled omponents of veloity is, on the erage, redued with inreasing axial distane from the rotor. From these observations, it seems questionable to alulate the blade loading, through blade element theory, as presented in setion B, however Fig. 15 shows a omparison between experiments (taken from [15]) and model for the axial omponent of veloity where the agreement is muh better. Also visible in Fig. 15, the rapid evolution of axial omponent of veloity at the tip and root of the atuator surfae raises questions as to the origin of these variations. This harateristi is typial of the atuator surfae representation : it was already observed in a previous work [1] and must be linked with the ourrene of hordwise vortiity along the atuator surfae. In the ase of the rotating atuator surfae, all solutions exhibit at the enter of the rotor an inrease in axial veloity above its free-stream value, or equivalently a positive indued veloity. Fig. 1 : axial omponent of of u as a funtion of azimuth for different radial loations, λ=8 These observed disagreements an be attributed to several fators, the first being the evident differene between the model and the reality of the experiment. Another important fator regards the effet of rotation on the blade aerodynamis whih an dramatially hange lift and drag harateristis, hene the irulation along a blade setion when ompared to D harateristis. Similar to a blokage effet that is not taken into aount by the model, the absene of solid boundary to model the blades of the rotor most probably influenes the numerial solution. In the entral part of the rotor, the indued blokage effet might even be stronger sine loal solidity /r is higher. The unertainties assoiated with the disretization (domain size and mesh, onvetion model, numerial dissipation) of the problem and the boundary onditions treatment are also a ause for the observed disrepanies RE&PQJ, Vol. 1, No.6, Marh 008

8 Fig. 13 : tangential omponent of u as a funtion of azimuth for different radial loations, λ=8 Fig. 15 : Radial distribution of axial veloity at the middle of the blade (θ=π/) 5. Conlusion To model lifting devies, whether that be a wing in translation or rotation, this paper presents the onept of Fig. 14 : radial omponent of u as a funtion of azimuth for different radial loations, λ=8 atuator surfae, defined as a porous moving surfae that arries veloity and pressure disontinuities A CFD model is used to model the flow that results from this singular surfae. Adaptations to the CFD models are presented and the ase of the TUDelft experimental rotor is thoroughly analyzed. Comparisons between numerial model output and experimental measurements are presented for the oeffiients of power and of thrust and for near-wake veloity measurements. Results are mitigated but still enouraging. Values of thrust oeffiient alulated by the model are loser to experiment than oeffiient of power and it was found that details of the flow struture inherent to vortial wake are well reprodued by the proposed model. These are good signs that the model should be appropriate to analyze wake evolution. Further work will study in more details the proposed model and improvements are still underway. It is notably expeted to ompare the proposed model with other models for wind turbine aerodynamis and to analyze other models of wind turbines RE&PQJ, Vol. 1, No.6, Marh 008

9 Referenes [1] Shrek S., Hand M., Fingersh L., Robinson M., Simms D., NREL Unsteady Aerodynamis Experiment Wind Tunnel Test : Flow Field Charaterization and Model Preditions, Proeedings of the European Wind Energy Conferene, Copenhagen, Denmark, pp.35-47, 001. [] Leishman J. G., Challenges in Modeling the Unsteady Aerodynamis of Wind Turbines, AIAA Paper , 00. [3] Van Kuik, G.A.M., The Edge Singularity at an Atuator Dis with a Constant Normal Load, AIAA Paper No , 003. [4] Shen W. Z., Mikkelsen R., Sorensen J N, Tip Loss Corretions for Wind Turbine Computations, Wind Energy, vol. 8, pp , 005. [5] Leler C., Masson C., Wind Turbine Performane Preditions using a Differential Atuator Disk Modeling, Proeedings of the onferene The Siene of making Torque from Wind, Delft University, Netherlands, april 004. [6] Leler C., Masson C., Toward Blade-Tip Vortex Simulation with an Atuator-Lifting Surfae Model, A Colletion of the 004 ASME Wind Energy Symposium Tehnial Papers, Reno, pp , January 004. [7] Rajagopalan, R.G., Fanui, J.B., Finite Differene Model for the Vertial Axis Wind Turbines, Journal of Propulsion and Power, Vol. 1, pp , [8] Masson C., Ammara I., Parashivoiu I., An Aerodynami Method for the Analysis of Isolated Horizontal-Axis Wind Turbines, International Journal of Rotating Mahinery, Vol. 3, pp. 1-3, [9] Sorensen, J.N., Myken, A., Unsteady Atuator Disk Model for Horizontal Axis Wind Turbines, Journal of Wind Engineering and Industrial Aerodynamis, 39, pp , 199. [10] Hansen, A.C., Butterfield, C.P., Aerodynamis of Horizontal-Axis Wind Turbines, Annual Review of Fluid Mehanis, Vol. 5, pp , [11] Leler C., Masson C., Wind Turbine Performane Preditions using a Differential Atuator Disk Modeling, Proeedings of the onferene The Siene of making Torque from Wind, Delft University, Netherlands, april 004. [1] Sibuet Watters C., Masson C., Reent Advanes in Modeling of Wind Turbine Wake Vortial Struture Using a Differential Atuator Disk Theory, Proeedings of the seond onferene The Siene of making Torque from Wind, Danish Tehnial University, Denmark, August 007. [13] Saabas H.J., Baliga B.R., Co-Loated Equal-Order Control-Volume Finite-Element Method for Multidimensional, Inompressible, Fluid Flow - Part I, Numerial Heat Transfer, Vol. 6B, pp , [14] Leler C., Masson C., Wind Turbine Performane Preditions using a differential atuator-lifting disk model, ASME Journal of Solar Energy Engineering, Vol. 17, N., pp , May 005. [15] Sant T., Ph.D. dissertation : Improving BEM-based Aerodynami Models in Wind Turbine Design Codes, TU Delft (Cum Laude), January 007. [16] Haans W., Sant T., van Kuik G.A.M., van Bussel G.J.W., HAWT Near-Wake Aerodynamis, Part I: Axial Flow Conditions, Wind Energy, Vol. 10, January 008. [17] Sheldahl R. E. and Klimas P. C., Aerodynami Charateristis of Seven Airfoil Setions Through 180 Degrees Angle of Attak for Use in Aerodynami Analysis of Vertial Axis Wind Turbines, SAND80-114, 1981, Sandia National Laboratories, NM. [18] Snel H., Review of Aerodynamis for Wind Turbines, Wind Energy, Vol. 6, 3, pp , 003 [19] Sørensen, N.N., CFD modelling of wind turbine aerodynamis. In: Wind turbine aerodynamis: A stateof-the-art. von Karman Institute for Fluid Dynamis leture series, Rhode Saint Genese (BE), 19-3 Mar RE&PQJ, Vol. 1, No.6, Marh 008